Averaging in cascade junctions with a~``wide'' transmission domain

T. P. Chechkina

Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 39 (2011), pp. 151-162.

Résumé

We consider a boundary-value problem for the Poisson equation in a dense multilevel junction consisting of a body of the junction and many periodically situated thin rectangles connected with the body by a transmission layer with a periodic boundary. It is supposed that the rectangles have finite lengths and the transmission layer has a small width that is much greater than the period. Nonhomogeneous Neumann boundary conditions are imposed upon the boundary of the transmission layer. An addition parameter of perturbation is included in these boundary conditions. Averaged problems are constructed and convergence theorems for solution and energy integrals are proved as a function of that parameter.

Bibliographie

[1] Bakhvalov N. S., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[2] Belyaev A. G., O singulyarnykh vozmuscheniyakh granichnykh zadach, Diss. k. f.-m. n., Moskva, 1990

[3] Belyaev A. G., Pyatnitskii A. L., Chechkin G. A., “Asimptoticheskoe povedenie resheniya kraevoi zadachi v perforirovannoi oblasti s ostsilliruyuschei granitsei”, Sib. mat. zhurn., 39:4 (1998), 730–754 | MR | Zbl

[4] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993, 461 pp. | MR | Zbl

[5] Kotlyarov V. P., Khruslov E. Ya., “O predelnom granichnom uslovii odnoi zadachi Neimana”, Teoriya funktsii, funkts. an. i ikh prilozh., 10, 1970, 83–96 | Zbl

[6] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973, 407 pp. | MR

[7] Marchenko V. A., Khruslov E. Ya., Kraevye zadachi v oblastyakh s melkozernistoi granitsei, Naukova dumka, Kiev, 1974, 279 pp. | MR

[8] Melnik T. A., “Userednenie ellipticheskikh uravnenii, opisyvayuschikh protsessy v silno neodnorodnykh tonkikh perforirovannykh oblastyakh s bystro izmenyayuscheisya tolschinoi”, Dokl. AN Ukrainy, 1991, no. 10, 15–19 | MR

[9] Melnik T. A., Vaschuk P. S., “Usrednenie kraevoi zadachi so smennym tipom granichnykh uslovii v gustom soedinenii”, Differ. uravn., 43:5 (2007), 677–685 | MR

[10] Melnik T. A., Nazarov S. A., “Asimptoticheskaya struktura spektra v zadache o garmonicheskikh kolebaniyakh stupitsy s tyazhelymi spitsami”, Dokl. RAN, 333:1 (1993), 13–15 | MR

[11] Melnik T. A., Nazarov S. A., “Asimptotika resheniya spektralnoi zadachi Neimana v oblasti tipa ‘`gustogo grebeshka’ ”, Tr. sem. im. I. G. Petrovskogo, 19, 1996, 138–174

[12] Melnik T. A., Nazarov S. A., “Asimptoticheskii analiz zadachi Neimana na soedinenii tela s tonkimi tyazhelymi sterzhnyami”, Algebra i analiz, 12:2 (2000), 188–238 | MR | Zbl

[13] Melnik T. A., Chechkin G. A., “Usrednenie kraevoi zadachi v gustom kaskadnom soedinenii”, Problemy mat. analiza, 37, 2008, 47–72 | MR

[14] Melnik T. A., Chechkin G. A., “Asimptoticheskii analiz kraevykh zadach v gustykh kaskadnykh soedineniyakh”, Dopovidi NAN Ukraïni, 2008, no. 9, 16–22 | MR

[15] Melnik T. A., Chechkin G. A., “Asimptoticheskii analiz kraevykh zadach v gustykh trëkhmernykh mnogourovnevykh soedineniyakh”, Mat. sb., 200:3 (2009), 49–74 | MR | Zbl

[16] Melnik T. A., Chechkin G. A., Chechkina T. P., “Teoremy skhodimosti dlya reshenii i integralov energii kraevoi zadachi s vozmuschennymi kraevymi usloviyami Neimana na granitsakh tonkikh sterzhnei v gustykh mnogourovnevykh soedineniyakh novogo tipa”, Problemy mat. analiza, 40, 2009, 113–131 | MR

[17] Nazarov S. A., “Soedineniya singulyarno vyrozhdayuschikhsya oblastei razlichnykh predelnykh razmernostei, I”, Tr. sem. im. I. G. Petrovskogo, 18, 1995, 1–78

[18] Nazarov S. A., “Soedineniya singulyarno vyrozhdayuschikhsya oblastei razlichnykh predelnykh razmernostei, II”, Tr. sem. im. I. G. Petrovskogo, 20, 2000, 155–196

[19] Nazarov S. A., Asimptoticheskaya teoriya tonkikh plastin i pryamougolnikov, Nauchnaya kniga, Novosibirsk, 2002

[20] Pyatnitskii A. L., Chechkin G. A., Shamaev A. S., Usrednenie. Metody i prilozheniya, Tamara Rozhkovskaya, Novosibirsk, 2007

[21] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988, 334 pp. | MR

[22] Sobolev S. L., Izbrannye voprosy teorii funktsionalnykh prostranstv i obobschennykh funktsii, Nauka, M., 1989, 255 pp. | MR

[23] Khruslov E. Ya., “O rezonansnykh yavleniyakh v odnoi zadache difraktsii”, Teoriya funktsii, funkts. an. i ikh prilozh., 6, 1968, 111–129 | Zbl

[24] Chechkin G. A., Chechkina T. P., “Ob usrednenii zadach v oblastyakh tipa ‘`infuzorii’ ”, Tr. sem. im. I. G. Petrovskogo, 23, 2003, 386–407 | MR | Zbl

[25] Chechkin G. A., Chechkina T. P., “Teorema usredneniya dlya zadach v oblastyakh tipa “infuzorii” s nesoglasovannoi strukturoi”, Differentsialnye uravneniya s chastnymi proizvodnymi, Sovr. matem. i ee prilozh., 2, In-t kibernetiki AN Gruzii, 2003, 139–154 | MR

[26] Amirat Y., Chechkin G. A., Gadyl'shin R. R., “Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary”, Zh. Vychisl. Mat. Mat. Fiz., 46:1 (2006), 102–115 | MR | Zbl

[27] Amirat Y., Chechkin G. A., Gadyl'shin R. R., “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues”, Appl. Anal., 86:7 (2007), 873–897 | DOI | MR | Zbl

[28] Amirat Y., Chechkin G. A., Gadyl'shin R. R., “Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary”, C. R. Mécanique, 336:9 (2008), 693–698 | DOI | Zbl

[29] Amirat Y., Chechkin G. A., Gadyl'shin R. R., “Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots”, Math. Methods Appl. Sci., 3:7 (2010), 811–830 | MR

[30] Bensoussan A., Lions J.-L., Papanicolau G., Asymptotic analysis for periodic structures, North-Holland Publishing Co., Amsterdam, 1978 | MR

[31] Blanchard D., Gaudiello A., Griso G., “Junction of a periodic family of elastic rods with 3D plate, I”, J. Math. Pures Appl. (9), 88:9 (2007), 1–33 | MR | Zbl

[32] Blanchard D., Gaudiello A., Griso G., “Junction of a periodic family of elastic rods with 3D plate, II”, J. Math. Pures Appl. (9), 88:9 (2007), 149–190 | MR | Zbl

[33] Blanchard D., Gaudiello A., Mel'nyk T. A., “Boundary homogenization and reduction of dimension in a Kirchhoff–Love plate”, SIAM J. Math. Anal., 39:6 (2008), 1764–1787 | DOI | MR | Zbl

[34] Blanchard D., Gaudiello A., Mossino J., “Highly oscillating boundaries and reduction of dimension: the critical case”, Anal. and Appl. (Singap.), 5 (2007), 137–163 | DOI | MR | Zbl

[35] Bouchitté G., Lidouh A., Suquet P., “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire et un corps rigide”, C. R. Acad. Sci. Paris. Sér. I, 313 (1991), 967–972 | MR | Zbl

[36] Chechkin G. A., Chechkina T. P., D'Apice C., De Maio U., Mel'nyk T. A., “Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission Zone”, Applicable Analysis, 88:10–11 (2009), 1543–1562 | DOI | MR | Zbl

[37] Chechkin G. A., Chechkina T. P., D'Apice C., De Maio U., Mel'nyk T. A., “Homogenization of 3D thick cascade junction with the random transmission zone periodic in one direction”, Russ. J. Math. Phys., 17:1 (2010), 35–55 | DOI | MR | Zbl

[38] Chechkin G. A., Cioranescu D., “Vibration of a Thin Plate with a “Rough” Surface”, Nonlinear partial differential equations and their applications, Elsevier, Amsterdam–London–New York–Tokyo, 2002, 147–169 | MR | Zbl

[39] Chechkin G. A., Friedman A., Piatnitski A. L., “The boundaryvalue problem in domains with very rapidly oscillating boundary”, J. Math. Anal. Appl., 231:1 (1999), 213–234 | DOI | MR | Zbl

[40] Cioranescu D., Saint Jean Paulin J., Homogenization of reticulated structures, Springer-Verlag, Berlin–New York, 1999 | MR | Zbl

[41] Dancer E. N., Daners D., “Domain perturbation for elliptic equations subject to Robin boundary conditions”, J. Differential Equations, 138:1 (1997), 86–132 | DOI | MR | Zbl

[42] D'Apice C., De Maio U., Mel'nyk T. A., “Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2”, Netw. Heterog. Media, 2 (2007), 255–277 | DOI | MR

[43] De Maio U., Durante T., Mel'nyk T. A., “Asymptotic approximation for the solution to the robin problem in a thick multi-level junction”, Math. Models Methods Appl. Sci., 15:12 (2005), 1897–1921 | DOI | MR | Zbl

[44] Durante T., Mel'nyk T. A., “Asymptotic analysis of a parabolic problem in a thick two-level junction”, Zh. Mat. Fiz. Anal. Geom., 3:3 (2007), 313–341 | MR | Zbl

[45] Durante T., Mel'nyk T. A., Vashchuk P. S., “Asymptotic aproximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction”, Nonlinear Oscil. (N.Y.), 9:3 (2006), 336–355 | MR

[46] Gaudiello A., “Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary”, Ricerche Mat., 43 (1994), 239–292 | MR | Zbl

[47] Jäger W., Mikelić A., “On the roughness-induced boundary conditions for an incompressible viscous flow”, J. Differential Equations, 170:1 (2001), 96–122 | DOI | MR

[48] Kazmerchuk Yu. A., Mel'nyk T. A., “Homogenization of the Signorini boundary-value problem in a thick plane junction”, Nonlinear oscil. (N.Y.), 12:1 (2009), 45–59 | DOI | MR

[49] Kohler W., Papanicolaou G., Varadhan S., “Boundary and interface problems in regions with very rough boundaries”, Multiple scattering and waves in random media, North-Holland Publishing Co., Amsterdam, 1981 | MR | Zbl

[50] Mel'nyk T. A., “Homogenization of the Poisson equation in a thick periodic junction”, Z. Anal. Anwendungen, 18:4 (1999), 953–975 | MR

[51] Mel'nyk T. A., “Homogenization of a singularly perturbed parabolic problem in a thick periodic junction of type 3:2:1”, Ukrainian Math. J., 52:11 (2000), 1737–1749 | DOI | MR

[52] Mel'nyk T. A., “Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multi-level junction”, Ukranian Math. J., 58:2 (2006), 220–243 | DOI | MR

[53] Mel'nyk T. A., “Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1”, Math. Models Methods Appl. Sci., 31:9 (2008), 1005–1027 | MR

[54] Nevard J., Keller J. B., “Homogenization of rough boundaries and interfaces”, SIAM J. Appl. Math., 57:6 (1997), 1660–1686 | DOI | MR | Zbl